2014-03-11

A common brain teaser is to connect a 3 by 3 grid of dots with four straight lines without picking up the pencil. After solving a 3x3, try to solve a 4x4, 5x5, etc. I have a proof that all of them have a solution.

Click read more to see the solutions and explanations:

Here's the solution for a basic 3x3 grid, start at either the top-left or bottom right dot:

Observant readers should realize that there is a pattern to these solutions that I draw. That base 3x3 solution is included in each of the subsequent solutions. That was the trick to solving the larger grids. When first figuring out the 4x4, I didn't realize the pattern until I got to the 5x5. Then, everything clicked and suddenly became easier.

As the grid width and length gets larger by one, you can think of it as another row and column of dots being added on two of the sides. Then, you overlay your previous solution on top, extend one of the lines, and it is easy to cover up a perpendicular row and column of dots with two lines.

Now that you know the trick, you should be able to solve the following 6x6 grid with 10 contiguous, straight lines:

A blank 6x6 grid of dots.

On the next page, I have a picture of the answer, and some more details on how I worked out the solutions for the first time.

In each of the solutions I've provided in this blog post, I kept the orientation of the 3x3 grid the same so that it would possibly be easier for readers to follow how each of the grids relate to each other.

For the 4x4 puzzle, first I tried just looking at it and trying to solve it many different ways in my head. When that didn't work, I drew one line from the top-left to bottom-right because I realized that I had to start somewhere, and by drawing the first line, it significantly reduced the difficulty of the problem. In my head, I drew a second line from the bottom-left to top-right and a different second line from the bottom-left to bottom-right. Then, I just tried other brute forcing methods, using the most logical lines first in order to cover the most dots per line and not repeat any dots that were already covered. Soon, I was able to arrive at the 4x4 solution shown on the last page.
0
Here's how some of my work looked like when I was trying to figure out the solutions for the first time:

Working out possible answers for connecting grid dots on a whiteboard.

The two drawings on the left-bottom is trying to figure out possible ways to draw the lines to cover up the most dots at once per line. If you can combine the two successfully, then that would be another solution. I also drew a 5x5 grid with four lines connecting all the outer dots, leaving the center 9 dots untouched. That's when I figure that I could use the same technique as solving the other puzzles.

And, for those special readers that made it this far, I have a secret (and another puzzle) for you. It is possible to complete each of these dot grid puzzles (3x3, 4x4, etc) with just a single line, with no twists or turns. Hint: Don't assume an Euclidean geometry. ;) If I get time, then I will write a post about the four different ways to solve the 3x3 grid.